Computer Graphics and Geometric Modelling -- a Hybrid Approach

Computer-Aided Design, 1986

they also lack application independency 8. In most mech-GEODERM, a microcomputer-based solid modeller, which anical design applications, there is a need to generate incorporates the parametric object model, is discussed. The mechanical parts having the same topology but different entity-relationship model, which is used to describe the dimensions. This necessitates the need to represent shape conceptual schema of the geometric database, is also pretopology and dimension at a conceptual level. sented. Three of the four modules of GEODERM, which In robot motion simulation, it is often necessary to have been implemented are described in some detail. They maintain relationships between object parts like the robot are the Solid Definition Language (SDL), the Solid Manipuarm and the base. These relationships should be easily lotion Language (SML) and the User-System Interface.

Computers & Industrial Engineering, 1988

1993

The parimod system is a Transputer based graphics system with an additional interactive solid modeling tool for fast rendering of arbitrary 3-dimensional scenes. It consists of an input tool, a calculation and an output unit which are independent of each other so that each of them is replaceable if changes in hard or software come through. The input unit is a X based solid modeling tool allowing the user to define scenes like an architect on his drawing board. With the massively parallel rendering tool, the user sees the defined scenes in various qualities on-line on the true color output device. The fast output is achieved by the implementation of different parallel strategies of well-known shading and rendering algorithms. The quick response time between changing and showing the different views of a scene makes parimod very popular in the application fields of architectural drawings or the visualization of molecules.

Computers & Graphics, 2004

The language of geometric algebra can be used in the development of computer graphics applications. This paper proposes a method to describe a 3D polygonal mesh model using a representation technique based on geometric algebra and the conformal model of the 3D Euclidean space. It describes also the stages necessary to develop an application that uses this formalism. The current application was used to validate the implementation of the main abstract operations characteristic to a geometric algebra computational environment (programming module GAP). The data structures that characterize this geometric algebra based modeling approach as well as the implementation of geometric algebra based methods for model visualization/transformation are developed in detail. The paper emphasizes the elegance and generality of the geometric algebra approach referring also to the necessary computational resources.

2007

A geometric model of an object – in most cases being a subset of the three dimensional space – can be used to better understand the object’s structure or behavior. Therefore data such as the geometry, the topology and other application specific data have to be represented by the model. With the help of a computer it is possible to manipulate, process or display these data. We will discuss different approaches for representing such an object: Volume based representations describe the object in a direct way, whereas boundary representations describe the object indirectly by specifying its boundary. A variety of different surface patches can be used to model the object’s boundary. For many applications it is sufficient to know only the boundary of an object. For special objects explicit or implicit mathematical representations can easily be given. An explicit representation is a map from a known parameter space for in- stance the unit cube to 3D-space. Implicit representations are equations or relations such as the set of zeros of a functional with three unknowns. These can be very efficient in special cases.

We propose a high-level, generative, textual representation for featurebased solid modeling, which we call Erep. We argue that such a representation should be independent of an underlying core solid modeler, and give some criteria it should satisfy. Such an Erep allows archiving geometric designs in a form that is both editable and translatable to any solid modeling system. Furthermore, the representation serves as a global schema by which to federate different modeling systems, and is extensible in a natural way to a representation from which to derive analysis representations and process plans. By federating with finite-element analysis packages, in particular, our approach offers closing the design-analysis feedback loop that previously required a manual link.

1995

This paper explores the use of visual operators for solids modeling. We focus on designing interfaces for free-form operators such as blends, sweeps, and deformations, because these operators have a large number of interacting parameters whose effects are often determined by an underlying parameterization. In this type of interactive modeling good solutions to the design problem have aesthetic as well as engineering components. Traditionally, interaction with the parameters of these operators has been through text editors, curve editors, or trial-and-error with a slider bar. Parametric values have been estimated from data, but not interactively. These parameters are usually one-or two-dimensional, but the operators themselves are intrinsically three-dimensional in that they are used to model surfaces visualized in 3D. The traditional textual style of interaction is tedious and interposes a level of abstraction between the parameters and the resulting surface. A 3D visual interface has the potential to reduce or eliminate these problems by combining parameters and representing them with a higherlevel visual tool. The visual tools we present not only speed up the process of determining good parameter values but also provide visual interactions that are either independent of the particular parameterizations or make explicit the effect of the parameterizations. Additionally, these tools can be manipulated in the same 3D space as the surfaces produced by the operators, supporting quick, interactive exploration of the large design space of these free-form operators. This paper discusses the difficulties in creating a coherent user interface for interactive modeling. To this end we present four principles for designing visual operators, using several free-form visual operators as concrete examples.

Programming and Computer Software, 2017

A ray-tracing algorithm for interactive visualization of very large and structurally complicated scenes presented in the constructive solid geometry (CSG) form is suggested. The algorithm is capable of visualizing such scenes in real time by using a graphic processor. As primitives, classical shapes and objects represented in an analytical form (in particular, second-order surfaces and implicit functions) are used. Unlike other similar algorithms, our algorithm produces the final image in a single pass and has no constraints on the maximum number of primitives and on the CSG tree depth. The key feature of the algorithm is a method for optimizing CSG models, which converts the input tree to an equivalent spatially coherent and well-balanced form (a completely balanced equivalent tree may not exist). The performance of visualization after applying the optimization technique is shown to depend on only the computational resource of the GPU (in contrast to multi-pass algorithms whose performance is restricted by memory capacity). It has been shown experimentally that our algorithm is capable of rendering CSG models consisting of more than a million CSG primitives with the tree depth up to 24.

The Visual Computer, 1997

We present a formal model for the boundary representation (B-rep) of polyhedral heterogeneous solids based on a theoretical model of graphic objects. The theoretical model is the graphic objects algebra, with which we manage solid modelling by enumeration, sweeping and CSG. We establish a formal mapping between graphic objects and solid models for heterogeneous objects, represented by CSG and B-rep. Using the algebra, we describe heterogeneous general polyhedra, manifold and nonmanifold, with and without holes. After a formal study of regularized operations in the algebra, which are a generalization of boolean regularized operations in solid modelling, we propose a B-rep method for managing graphic objects described by their boundaries. This facilitates an abstract uniform treatment of the most important solid modelling methods.

ACM Transactions on Graphics, 1995

This article presents a functional programming approach to geometric design with embedded polyhedral complexes. Its main goals are to show the expressive power of the language as well as its usefulness for geometric design. The language, named PIASM (the ProgrammingIAnguage for Solid Modeling), introduces a very high level approach tn "constructive" or "generative" modeling. Geometrical objects are generated by evaluating some suitable language expressions. Because generating expressions can be easily combined, the language also extends the standard variational geometry approach by supporting classes of geometric objects with varying topology and shape. The design language PLASM can be roughly considered as a geometry-oriented extension of a subset of the functional language FL. The language takes a dimension-independent approach to geometry representation and algorithms. In particular it implements an algebraic calculus over embedded polyhedra of any dimension. The generated objects are always geometrically consistent because the validity of geometry is guaranteed at a syntactical level. Such an approach allows one to use a representation scheme which is weaker than those usually adopted in solid modelers, thus encompassing a broader geometric domain, which contains solids, surfaces, and wire-frames, as well as higher-dimensional objects. Geometric Programming . 267 Son of man,. . . . show them the design and plan of the Temple, its exits and entrances, its shape, how all of it is arranged, the entire design and all its principles, Give them all this in writing so that they can see and take note of its design and the wa.v it is all arranged and carry it out.

Computer-Aided Design, 2008

2004

A geometric model of an object -in most cases being a subset of the three dimensional space -can be used to better understand the object's structure or behaviour. Therefore data such as the geometry, the topology and other application specific data have to be represented by the model. With the help of a computer it is possible to manipulate, process or display these data. We will discuss different approaches for representing such an object: Volume based representations describe the object in a direct way, whereas boundary representations describe the object indirectly by specifying its boundary. A variety of different surface patches can be used to model the object's boundary. For many applications it is sufficient to know only the boundary of an object. For special objects explicit or implicit mathematical representations can easily be given. An explicit representation is a map from a known parameter space e.g. the unit cube to 3D-space. Implicit representations are equations or relations such as the set of zeros of a functional with three unknowns. These can be very efficient in special cases. As an example of volume based representations we will give a brief overview of the voxel representation. We also show how the boundary of complex objects can be assembled by simpler parts e.g. surface patches. These come in a variety of forms: planar polygons, parametric surfaces, especially spline surfaces and trimmed surfaces, multiresolutionally represented surfaces, e.g. wavelet-based surfaces. In a boundary representation only the boundary of a solid is described. This is usually done by describing the boundary as a collection of surface patches attached to each other at outer edges. Simple objects constructed via any of the methods above can be joined to build more complex objects via Boolean operators (constructive solid geometry, CSG). Constructing an object one has to assure that the object is in agreement with the topological requirements of the modeling system. Notoriously difficult problems are caused by the fact that most modeling systems can compute surface intersections only with a limited precision. This yields numerical results that may finally cause major errors e.g. topologically contradictory conclusions. The rather new method of "Medial Modeling" is also presented. Here an object is described by its medial axis and an associated radius function. The medial axis itself is a collection of lower dimensional objects, i.e. for a 3D-solid a set of points, curves and surface patches. This medial modeling concept developed at the Welfenlab yields a very intuitive user interface useful for solid modeling, and also gives a natural meshing of the solid for FEM computations. Additional attributes can be attached to an object, i.e. attributes of physical origin or logical attributes. Physical attributes include photometric, haptical and other material properties, such as elasticity or roughness. Physical attributes are often specified by textures, i.e. functions that relate surface points to certain quantities of the attribute. The most common use for these are photometric textures, although they can also be used for roughness etc. Logical attributes relate the object to its (data-)environment. They can e.g. group objects which are somehow related, or they can associate scripts to the object.

— For the realization of heterogeneous objects, CAD models are the prerequisites for the downstream applications of analysis, fabrication and visualization. Thus, the model should be able to communicate and share the data among analysis, fabrication and visualization modules. System level research and development for successful implementation of such CAD models are limited. The present work unfold the system requirements and focuses on developing an algorithm and data structure for efficient data generation and effective data transfer for modeling, visualization, analysis, fabrication and material information retrieval of heterogeneous objects. Step-wise algorithm is represented for geometric and material modeling in the heterogeneous object domain. Developed data structure illustrates the ways of data interaction between different processes starting from solid modeling to analysis and fabrication of heterogeneous objects. An example of heterogeneous object has been demonstrated to validate the developed algorithm and data structure.

Proceedings of the 1986 workshop on Interactive 3D graphics - SI3D '86, 1987

The success of solid modelling in industrial design depends on facilities for specifying and editing parameterized models of solids through user-friendly interaction with a graphical front-end. Systems based on a dual representation, which combines Constructive Solid Geometry (CSG) and Boundary representation (BRep), seem most suitable for modelling mechanical parts. Typically they accept a CSG-compatible input (Boolean combinations of solid primitives) and offer facilities for parameterizing and editing part definitions. The user need not specify the topology of the boundary, but often has to solve three-dimensional trigonometric problems to compute the parameters of rigid motions that specify the positions of primitive solids. A front-end that automatically converts graphical input into rigid motions may be easily combined with boundary-oriented input, but its integration in dual systems usually complicates the editing process and limits the possibilities of parameterizing solid definitions. This report proposes a solution based on three main ideas: (1) enhance the semantics of CSG representations with rigid motions that operate on arbitrary collections of sub-solids regardless of their position in the CSG tree, (2) store rigid motions in terms of unevaluated constraints on graphically selected boundary features, (3) evaluate constraints independently, one at a time, in user-specified order. The third idea offers an alternative to known approaches, which convert all constraints into a large system of simultaneous equations to be solved by iterative numerical methods. The resulting front-end is inadequate for solving problems where multiple constraints must be met simultaneously, but provides a powerful tool for specifying and interactively editing parameterized models of mechanical parts and mechanisms. The order in which constraints are evaluated may also be used as a language for specifying the sequence of assembly and setup operations. An implementation under way is based on the interpreter of a new object oriented programming language, enhanced with geometric classes. Constraint evaluation results in the activation of methods which compute rigid motions from surface information. The set of available methods may be extended by the users, and methods may be integrated in higher level functions whose algorithmic nature simplifies the treatment of degenerate cases. Graphic interaction is provided through a geometrical engine which lets the user manipulate shaded images produced efficiently from the CSG representation of solid models.

IEEE Computer Graphics and Applications, 1985

Standardization of graphics program packages, beginning about 10 years ago and culminating as an international standard,1-4 was based on the assumption that modeling and drawing (viewing) should be separate. At that time the output pipeline, with windowing and graphical output primitives, was well understood. Today, in many experimental systems5-10 a geometric model can be visualized in different ways: as a wireframe, a color raster picture, or a realistic shaded image. Problems arose on the input side, however. The proposed input primitives were soon found to be much too primitive and to have no direct correspondence with the output primitive functions.11 Also it was realized that input actions often refer to the existing picture on the screen (the "pick" device), which of course must then be stored. This data structure, called segment storage, is essentially a model of the picture.